# Permeable Breakwaters Performance Modeling: A Comparative Study of Machine Learning Techniques

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}= 0.95, OBJ function = 0.0273), and similarly, the exponential GPR model in predicting the transmission coefficient (R

^{2}= 0.98, OBJ function = 0.0267) showed the best performance and the highest correlation with the actual records and can further be used as a reference for engineers in practical work. Also, the sensitivity analysis of the proposed models determined that the relative height parameter of the rockfill material has the greatest contribution to the introduced breakwater behavior.

## 1. Introduction

## 2. Experimental Set-Up

#### 2.1. Physical Model Scale

#### 2.2. Test Facility

#### 2.3. Physical Model Details

## 3. Methodology

#### 3.1. Linear Regression Model

#### 3.2. Regression Tree

_{w}(t) is the weighted number of records in node t, W

_{i}is the weighting field value for the record i (if any), f

_{i}is the repeat field value (if any), У

_{i}is the target field value, and $\mathrm{\u0233}\left(\mathrm{t}\right)$ is the mean value of the responding variable at node t.

_{R}) is the sum of the squares of the right child node, R(t

_{L}) is the sum of the squares of the left child node, and the division of s is also chosen to maximize the value of Q(s,t).

#### 3.3. Multilayer Perceptron Artificial Neural Network

_{ij}) [23]. The Y output is obtained by transferring the sum of the previous outputs and mapping by an activation function. Figure 3 basically illustrates the multilayer perceptron neural network. In this figure, X

_{i}represents the inputs, and b/B represents the bias between the different layers. For complex and nonlinear problems, the hyperbolic tangent function or the sigmoid (or log-sigmoid) function can be used.

#### 3.4. Support Vector Machine

_{1},y

_{1}),…(x

_{n},y

_{n}), x ϵ Rn, y ϵ r, where x is input, y is output, R

_{n}is the n-dimensional vector space, and r is the one-dimensional vector space. The ε-insensitive loss of function can be expressed as follows:

_{i},x

_{j}) is a kernel function. Some common kernels such as homogeneous and non-homogeneous polynomials, radial basis function, and Gaussian function. In the present study, Gaussian, linear, quadratic, and cubic kernel functions are used to investigate the model.

#### 3.5. Gaussian Process Regression

_{i}, y

_{i}); i = 1, 2, …, n} such that x

_{i}∈ ℝ

^{d}and y

_{i}∈ ℝ are derived from an unknown distribution. A GPR model with respect to the new input vector x

_{new}and the training data supports the prediction of the variable response value of y

_{new}. A linear regression model is as follows:

^{T}β + ε

^{2}and the coefficient β are calculated from the data. A GPR model describes the response by introducing latent variables from a Gaussian process, f(x

_{i}), i = 1,2, …, n, and the explicit basic functions h. The covariance function of the hidden variables introduces smoothness of the response, and the base functions transfer the inputs of X to a p-dimensional feature space. A Gaussian process is a set of random variables such that any finite number of them, have a Gaussian distribution. If {f(x),x∈ℝd} is a Gaussian process and n has observations x

_{1}, x

_{2}, …, x

_{n}, the distribution of random variables f (x

_{1}), f (x

_{2}), …, f (x

_{n}) is also Gaussian. A Gaussian process is defined by the mean function m(x) and the covariance function k (x, x′). That is, if {f(x),x∈ℝd} is a Gaussian process, then E(f(x)) = m(x) and Cov[f(x),f(x′)] = E[{f(x)−m(x)}{f(x′)−m(x′)}] = k(x,x′). Now consider the following model:

^{T}β + F(x)

_{i}) is introduced for each xi observation, which makes the GPR model non-parametric [27].

#### 3.6. Genetic Programming

## 4. Modeling of Reflection and Transmission Coefficients

_{t}and C

_{r}, respectively, are transmission and reflection coefficients, B is the width of breakwater chamber (distance between front and back panel), h is the depth of water, d is the height of rockfill materials, L

_{p}is wavelength associated with peak period (T

_{p}) wave spectrum, H

_{s}significant wave height of wave spectrum, K is wave number or angular frequency (is equal to $\frac{2\mathsf{\pi}}{{\mathrm{L}}_{\mathrm{p}}}$), and p is the permeability of the back wall, defined as the ratio of permeable wall openings to the entire vertical surface of the wall from the water surface to the bed of the flume. Previous studies on Jarlan’s multi-layer breakwaters without the use of rockfill materials have suggested that the presence of a back wall with less porosity compared to the front wall will improve the performance of the breakwater [31,32]. In some other studies on multi-layer permeable breakwater, it has been shown that if the back wall has more than 40% porosity, it is ineffective. Given the fact that the introduced breakwater behaves differently from the previous breakwaters due to the presence of rockfill material in its core, this study seeks to investigate the effect of back wall porosity on suggested breakwater [33,34].

#### 4.1. Data and Preprocessing

_{p}), wave steepness (H

_{s}/L

_{p}), wave number multiplied by water depth (kh), and relative wave height in terms of rockfill height (H

_{s}/d). The primary purpose of using dimensionless parameters is to apply the results of a smaller-scale physical model to real work. Selected parameters have been considered and applied previously in several studies related to permeable breakwater [3,4,9] and have specific concepts in coastal engineering. For example, investigating the effect of H

_{s}/L

_{p}and kh on C

_{t}and C

_{r}is essential to understand the hydrodynamic characteristics of the present breakwater for coastal and deepwater regions. Also, investigating the effect of B/L

_{p}, $\mathrm{d}/\mathrm{h},\mathrm{B}/\mathrm{Lp}$, H

_{s}/d, and P on C

_{t}and C

_{r}is required to select the appropriate and optimized structure configuration.

_{c}groups or folds of approximately equal size. The first fold is treated as a validation set (testing), the modeling is fitted to the remaining (K

_{c}– 1) folds, and this partition is applied until each fold is tested one time and will continue (K

_{c}− 1) times to fit the model [35].

_{c}may lead to a false idea of the model skill level, such as having a high variance score (which may vary greatly based on the data used to fit the model) or high bias (which may be due to overestimating the model’s skills). K

_{c}is usually 5 or 10, but there is no formal rule. As K

_{c}increases, the difference between training datasets and subsequent iterations subsets becomes smaller. As the difference decreases, the bias becomes smaller. In this study, considering the number of data, k is equal to 8 (28). As a result, from a total of 360 sets of lab results, 315 datasets in 7-fold were used for the training process, and 45 were used in each step for model testing.

#### 4.2. Performance Measurement

- (1)
- Provide the best correlation expressed by the correlation coefficient
- (2)
- Provide the least error represented by the RMSE index

## 5. Results and Comparison

#### 5.1. Comparison of Models and Model Selection

#### 5.2. Parameter Settings of Selected Models

#### 5.3. Selected Models

#### 5.4. Sensitivity Study

_{p}= 0.5 + 0.5n) is another effective parameter in the reflection coefficient which is reflected in the B/L

_{p}parameter.

## 6. Conclusions

_{p}) parameters, and the back-wall porosity (P) parameter is the least important in this model. In the case of transmission coefficient modeling, the results show the sensitivity of the proposed model to the relative rockfill height (d/h) and the relative chamber width (B/h). Similarly, the model has the least sensitivity to the P parameter. Therefore, it has been concluded the wave hydrodynamic parameters (reflection and transmission coefficients) depend strongly on the relative rockfill height. In general, the reflection coefficient increases with the relative rockfill height (d/h), and conversely, the transmission coefficient decreases with an increasing relative rockfill height (d/h).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Error and Statistical Indicators Functions

_{i}and t

_{i}are, respectively, measured and predicted C

_{r}or C

_{t}values for the ith output, $\mathrm{h}\xaf$ and $\mathrm{t}\xaf$ are the average of the measured and predicted outputs, and n is the total number of tests.

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**Figure 2.**Details of wave flume, model position, and wave probes locations; (

**a**) Passive wave absorber; (

**b**) Model setup; (

**c**) Wave probes; (

**d**) Wave generating system

**Figure 4.**Typical genetic programming flowchart [27].

**Figure 5.**Measured versus predicted values for the top four models; (

**A**–

**D**) reflection prediction model and, (

**E**,

**F**) transmission prediction model.

**Figure 7.**The error distribution of predicted and actual values for (

**A**) Reflection Coefficient parameter (

**B**) Transmission Coefficient parameter.

Symbol | Definition | Range | Standard Deviation. |
---|---|---|---|

Lp | Spectrum wave length (cm) | 0.85–5.15 | 1.16 |

Hs | Spectrum significant wave (cm) | 2.91–14.20 | 3.21 |

d | Rockfill Height (cm) | 30–56 | 9.63 |

B | Chamber width (cm) | 16–80 | 22.63 |

Parameter | Definition | Range | Average | Standard Deviation |
---|---|---|---|---|

P | Back wall porosity (%) | 30–50 | 40 | 10 |

B/h | Relative chamber width | 0.4–2 | 1.2 | 0.57 |

d/h | Relative rockfill height | 0.75–1.4 | 1.09 | 0.24 |

B/L_{p} | Relative chamber width in terms of wavelength | 0.03–0.86 | 0.19 | 0.15 |

H_{s}/d | Relative wave height in terms of rockfill height | 0.05–0.45 | 0.18 | 0.09 |

Kh | The wavenumber multiplies the water depth | 0.49–2.96 | 1.01 | 0.57 |

H_{s}/L_{p} | Wave steepness | 0.01–0.06 | 0.03 | 0.01 |

C_{r} | Reflection coefficient | 0.16–0.54 | 0.37 | 0.09 |

C_{t} | Transmission coefficient | 0.11–0.77 | 0.44 | 0.1 |

ML Regressor | Reflection Coefficient | |||||
---|---|---|---|---|---|---|

E | Relative Error (RE) | Scatter Index (SI (%)) | RMSE | R Squared | OBJ Function (ρ) | |

Linear Regression | 0.8363 | 0.9536 | 9.7 | 0.036794 | 0.83 | 0.0520 |

Interactions Linear | 0.9348 | 0.9970 | 6.1 | 0.025197 | 0.92 | 0.0348 |

Robust Linear | 0.8360 | 0.9821 | 9.7 | 0.036859 | 0.83 | 0.0521 |

Stepwise Linear | 0.9259 | 0.9957 | 6.5 | 0.025446 | 0.92 | 0.0351 |

Fine Tree | 0.9751 | 0.9979 | 3.8 | 0.025985 | 0.92 | 0.0358 |

Medium Tree | 0.9494 | 0.9967 | 5.4 | 0.029467 | 0.89 | 0.0410 |

Coarse Tree | 0.8825 | 0.9909 | 8.2 | 0.035110 | 0.85 | 0.0494 |

Linear SVM | 0.8355 | 0.9886 | 9.7 | 0.036758 | 0.83 | 0.0520 |

Quadratic SVM | 0.9476 | 0.9960 | 5.5 | 0.023015 | 0.93 | 0.0317 |

Cubic SVM | 0.9679 | 0.9977 | 4.3 | 0.022148 | 0.94 | 0.0304 |

Medium Gaussian SVM | 0.9712 | 0.9979 | 4.1 | 0.021394 | 0.94 | 0.0294 |

Coarse Gaussian SVM | 0.8647 | 0.9899 | 8.8 | 0.033973 | 0.86 | 0.0476 |

Boosted Trees | 0.9358 | 0.9954 | 6.1 | 0.026707 | 0.91 | 0.0369 |

Bagged Trees | 0.9582 | 0.9969 | 4.9 | 0.024971 | 0.92 | 0.0344 |

Squared exponential GPR | 0.9730 | 0.9981 | 3.9 | 0.020357 | 0.95 | 0.0279 |

Matern 5/2 GPR | 0.9797 | 0.9985 | 3.4 | 0.020222 | 0.95 | 0.0277 |

Exponential GPR | 0.9999 | 0.9999 | 0.014 | 0.019949 | 0.95 | 0.0273 |

Rational Quadratic GPR | 0.9753 | 0.9982 | 3.8 | 0.020366 | 0.95 | 0.0279 |

GP | 0.9207 | 0.9940 | 6.7 | 0.025117 | 0.92 | 0.0346 |

ANN | 0.9100 | 0.9935 | 7.2 | 0.026791 | 0.91 | 0.0370 |

ML Regressor | Transmission Coefficient | |||||
---|---|---|---|---|---|---|

E | Relative Error (RE) | Scatter Index (SI (%)) | RMSE | R Squared | OBJ Function (ρ) | |

Linear Regression | 0.7960 | 0.9404 | 15.10 | 0.069152 | 0.78 | 0.0835 |

Interactions Linear | 0.9448 | 0.9856 | 7.85 | 0.039621 | 0.93 | 0.0458 |

Robust Linear | 0.7826 | 0.9356 | 15.58 | 0.071068 | 0.77 | 0.0860 |

Stepwise Linear | 0.9423 | 0.9849 | 8.02 | 0.038990 | 0.93 | 0.0451 |

Fine Tree | 0.9810 | 0.9951 | 4.60 | 0.038005 | 0.93 | 0.0440 |

Medium Tree | 0.9489 | 0.9867 | 7.55 | 0.043723 | 0.91 | 0.0509 |

Coarse Tree | 0.8740 | 0.9654 | 11.86 | 0.065657 | 0.81 | 0.0785 |

Linear SVM | 0.7866 | 0.9357 | 15.44 | 0.070599 | 0.78 | 0.0852 |

Quadratic SVM | 0.9584 | 0.9893 | 6.82 | 0.034423 | 0.95 | 0.0396 |

Cubic SVM | 0.9806 | 0.9950 | 4.65 | 0.027994 | 0.96 | 0.0321 |

Medium Gaussian SVM | 0.9804 | 0.9948 | 4.67 | 0.033301 | 0.95 | 0.0383 |

Coarse Gaussian SVM | 0.8351 | 0.9487 | 13.57 | 0.064471 | 0.81 | 0.0771 |

Boosted Trees | 0.9608 | 0.9894 | 6.61 | 0.034954 | 0.95 | 0.0402 |

Bagged Trees | 0.9705 | 0.9919 | 5.74 | 0.037562 | 0.94 | 0.0433 |

Squared exponential GPR | 0.9897 | 0.9974 | 3.38 | 0.024612 | 0.97 | 0.0282 |

Matern 5/2 GPR | 0.9945 | 0.9986 | 2.47 | 0.024526 | 0.97 | 0.0281 |

Exponential GPR | 0.9999 | 0.9999 | 0.02 | 0.023402 | 0.98 | 0.0267 |

Rational Quadratic GPR | 0.9931 | 0.9982 | 2.76 | 0.024767 | 0.97 | 0.0284 |

GP | 0.9195 | 0.9784 | 9.48 | 0.042169 | 0.94 | 0.0487 |

ANN | 0.9383 | 0.9838 | 8.3 | 0.036900 | 0.94 | 0.0426 |

**Table 5.**Parameters setting of the exponential GPR in the modeling of reflection and transmission coefficients.

Target | Parameter | Value |
---|---|---|

Reflection coefficient | ||

Kernel Function | Exponential | |

Kernel Scale | 1.664473 | |

Basic Function | Constant | |

Signal Standard Deviation | 0.063185 | |

sigma | 0.063185 | |

Transmission coefficient | ||

Kernel Function | Exponential | |

Kernel Scale | 1.664473 | |

Basic Function | Constant | |

Signal Standard Deviation | 0.105257 | |

sigma | 0.105257 |

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## Share and Cite

**MDPI and ACS Style**

Gandomi, M.; Dolatshahi Pirooz, M.; Varjavand, I.; Nikoo, M.R.
Permeable Breakwaters Performance Modeling: A Comparative Study of Machine Learning Techniques. *Remote Sens.* **2020**, *12*, 1856.
https://doi.org/10.3390/rs12111856

**AMA Style**

Gandomi M, Dolatshahi Pirooz M, Varjavand I, Nikoo MR.
Permeable Breakwaters Performance Modeling: A Comparative Study of Machine Learning Techniques. *Remote Sensing*. 2020; 12(11):1856.
https://doi.org/10.3390/rs12111856

**Chicago/Turabian Style**

Gandomi, Mostafa, Moharram Dolatshahi Pirooz, Iman Varjavand, and Mohammad Reza Nikoo.
2020. "Permeable Breakwaters Performance Modeling: A Comparative Study of Machine Learning Techniques" *Remote Sensing* 12, no. 11: 1856.
https://doi.org/10.3390/rs12111856